Sidon pair encoding

Integrate this at the pair-feature stage of a graph neural network or transformer with symmetric entity interactions. First choose K entity types and construct an integer set A inside [0,N] using a greedy B_{2,\Delta} procedure. Initialize A and the occupied sum list as empty; scan candidate integers x; reject x if |x-a|<\Delta for any existing code a, if any new sum x+a or 2x lies within distance \Delta of an occupied sum, or if two newly generated sums are closer than \Delta; otherwise accept x and add its new sums. Compact pseudocode is: A=[]; sums=[]; for x in candidates: new=[x+a for a in A]+[2*x]; if min_distance(x,A)>=Delta and min_distance(new,sums)>=Delta and min_pairwise_distance(new)>=Delta: A.append(x); sums.extend(new); if len(A)==K: break. Store the resulting codes as a non-trainable vector a. For each unordered pair, compute s=a[i]+a[j], normalize by S=max_{u,v}|a[u]+a[v]|, and concatenate it to the usual edge or attention feature before an MLP. In attention, use logit_ij = q_i dot k_j / sqrt(d) + w^T MLP(s_ij/S). Optionally quantize s to int8 or add bounded noise and decode to the nearest valid sum; the B_{2,Delta} guarantee gives tolerance below Delta/2. The mathematical quantities are the injective pair-sum map and its separation margin; the code range N, MLP weights, and normalization are empirical design choices. Start with a 20-node synthetic graph task where labels depend on arbitrary unordered node-type pairs. Compare Sidon codes, random integer codes, and learned scalar codes at equal pair-feature width, with and without int8 quantization. Measure collision rate, pair-label accuracy, robustness under additive noise, code range, and memory. Success is zero collisions by construction and higher accuracy or lower pair-feature memory at equal model size.